Question: Find the distance between the points $(1,1)$ and $(4,7)$. Express your answer in simplest radical form.
Solution: The horizontal separation between $(1,1)$ and $(4,7)$ is $4-1=3$ units.  The vertical separation between the points is $7-1=6$ units.  Therefore, the segment whose endpoints are (1,1) and (4,7) is the hypotenuse of a right triangle whose legs measure 3 units and 6 units.  By the Pythagorean, the length of this segment is $\sqrt{3^2+6^2}=3\sqrt{1^2+2^2}=\boxed{3\sqrt{5}}$.